Seminar on low-dimensional mathematics Moscow-Petersburg
October 14, 2005, PDMI, room 311, 16:00--17:45.

Ricardo Uribe-Vargas (College de France)
"Topology of Dynamical Systems in Finite Groups and Number Theory"

Abstract:
We study and develop a very new object introduced by
V.I.~Arnold: a "monad" is a triple consisting of a finite set,
a map from that finite set to itself and the "monad graph"
which is the directed graph whose vertices are the elements of the
finite set and whose arrows lead each
vertex to its image (by the map). We consider the case in
which the finite set entering in the monad definition is a finite
group $G$ and the map $f:G\rightarrow G$ is the "Frobenius map"
$f_k:x\mapsto x^k$, for some $k\in \Z$. We study
the Frobenius dynamical system defined by the iteration of the
monad $f_k$, and also study the combinatorics and topology (i.e. the
discrete invariants) of the monad graph. Our study provides useful
information about several structures on the group associated to the monad
graph. So, for example, we obtain several properties of the quadratic
residues of finite commutative groups in terms of the graph of the
Frobenius monad $f_2:x\mapsto x^2$.

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